A global optimization method is used to predict the minimum energy structure of small protein-like molecules. This method begins by collecting a large number of molecular conformations, each obtained by finding a local minimum of a potential energy function from a random starting point. The information from these conformers is then used to form a convex quadratic global underestimating function (CGU) for the potential energy of all known conformers. This underestimator is an Ll approximation to all known local minima, and is obtained by a linear programming formulation and solution. The minimum of this estimator is used to predict the global minimum for the function, allowing a localized conformer search to be performed based on the predicted minimum. The new set of conformers generated by the localized search serves as the basis for another quadratic underestimation step in an iterative algorithm. Computational results obtained from this "CGU" method applied to actual protein sequences using a detailed polypeptide model and a differentiable form of the Sun/Thomas/Dill potential energy function are available. This potential function accounts for steric repulsion, hydrophobic attraction and ~~w pair restrictions imposed by the so called Ramachandran maps. Furthermore, it is easily augmented to accommodate additional known data such as the existence of disulphide bridges and any other a priori distance data. The Ramachandran data is modeled by a continuous penalty term in the potential function, thereby permitting the use of continuous minimization techniques.